This book is entirely devoted to the continuation (i.e. homotopy) method for solving the equation f(x) = 0. According to this method, the function f is embedded in a one-parameter family of equations f(x,λ) = 0, where f(.) = f(.,1) and a solution of f(x,0) = 0 is known. The first chapter is introductory and covers the necessary facts from functional analysis. In chapter 2, properties of general and special (e.g. linear) deformations of f are studied in the finite-dimensional case. The converse problem, i.e. the existence of a convenient deformation, is also investigated. In chapter 3, some results of the second chapter are extended to infinite dimensions. The Conley index is introduced in chapter 4 and its homotopy invariance is proved. The last chapter contains applications of the homotopy method to proofs of various classical inequalities, properties of extremals in nonlinear programming and calculus of variations, optimal control and bifurcation of critical points. The book is carefully written and it can be read by graduate students. Physicists and engineers who use variational methods will also find here a good source of information.